Department of Economics Working Paper University of Massachusetts Amherst Influence Networks and Public Goods Ináuence Networks and Public Goods

We consider a model of local public goods in a random network context. The ináuence network determines (exogenously) who observes whom every period and comprises a wide array of options depending on the degree distribution and the in/out-degree correlations. We show that there exists a unique equilibrium level of public good provision and compare it with the e¢cient level. We derive further insights for this problem by performing a comparative statics analysis.


Introduction
The study of networks has been of signiÖcant importance in diverse academic Öelds such as sociology, physics and computer science (see, e.g., Wasserman and Faust 1994;Newman 2003, and the long list of references cited therein). The last two decades have witnessed how numerous phenomena of economic relevance have also been studied using the paradigm of networks.
Instances are network formation (e.g., Jackson (Elliot et al. 2014) and microÖnance credits (Benerjee et al. 2013). 1 We consider a model of local public goods in a random network context. There are many socioeconomic situations which can be described as a local public good, that is, a good that is non-rival and non-excludible among neighbors in a relevant network. Some examples are innovations among collaborating Örms, complementarity of skills within social contacts, the provision of an open source product (e.g., software) or information in the Internet (e.g., websites or blogs). We shall be concerned with directed networks, i.e., networks in which the beneÖts from interacting with an agent providing the public good is only one way. We deÖne the outdegree (observability) of an agents as the number of agents she observes, whereas her in-degree (visibility) indicates how many agents observe this agent. The correlation between agentís outdegree and in-degree might depend on the application. For instance, in friendship networks, this correlation tends to be high because friendship is mostly bidirectional, whereas for other types of networks such as the WWW this correlation could potentially be lower.
As standard in this literature, we assume that the network is exogenously given in order to isolate the decision to contribute to the local public good from the network formation issue. We depart, however, from the Öxed network approach and consider random networks instead (see Agents have to decide whether to invest, or not, in the public good. If they enjoy the public good -either because they have invested in it or because they free ride on some other agent in the population that has done so-they obtain some beneÖts. This determines a game known as the best-shot game in which an agent has incentives to invest in the public good only if no other agent observed by her has already done so. The static version of this model on a Öxed (directed) underlying network structure has some limitations. On the one hand, for most networks structures there will be a large set of equilibrium outcomes and therefore we would incur in multiplicity problems. On the other hand, for some simple networks an equilibrium, in pure strategies, will not exist. We therefore propose and focus on an alternative approach based on a dynamics ináuence process which leads to a unique equilibrium prediction. In particular, there exists a unique globally stable outcome (fraction of individuals investing in the public good) of this dynamics characterized by the (out-)degree distribution and the in/out-degree correlation. Thus, some comparative statics results can be provided.
There are two outcomes of interest: (1) the fraction of public good providers in equilibrium, and (2) the fraction of links that reach a public good provider in equilibrium. The comparative statics results lead to the following conclusions. On the one hand, for any given out-degree distribution, an increase in the in/out-degree correlation increases measure (1) and decreases measure (2). On the other hand, for any in/out-degree correlation, if the network is su¢ciently dense, a First Order Stochastic Dominance shift (Mean Preserving Spread) of the out-degree distribution decreases (increases) measure (2).
We partially compare the e¢cient and equilibrium outcomes. We Önd that in the case of an homogeneous population in equilibrium there is underprovision of the public good. We also show that if we allow for some heterogeneity regarding out-degrees the e¢cient and equilibrium outcomes provide opposite results; the probability of contributing to the public good increases with respect to out-degree in equilibrium, whereas it decreases in the e¢cient state.
Public goods in a network context was Örst analyzed in the seminar paper by BramoullÈ on games of strategic substitutes on networks with linear best-reply functions which has recently been extended to non-linear settings by Allouch (2015). We contribute to this vast literature by analyzing local public goods in a random network context. The rest of the paper is organized as follows. In Section 2 we describe the model. In Section 3 we characterize the equilibrium. The comparative statics of the equilibrium outcome are discussed in Section 4. In Section 5 we provide some results on e¢ciency. Finally, in Section 6 we conclude.

The model
Before presenting the full speciÖcation of the model, let us start with the static benchmark case which will be useful to motivate our dynamic approach.

The static model
Let us consider a population with N = f1; :::; ng agents, where n is su¢ciently large. Agents observe each other due to what we describe as an ináuence network, which is exogenously given.
Let N i # N be the set of agents observed by i, where d i , the out-degree, is simply the cardinality of N i . The network is directed. That is, if an agent i observes (or is ináuenced by) j this does not imply that j observes i. For instance, an agent can observe the webpage of another agent but not vice versa. Each agent i 2 N chooses an action indicating whether or not to provide a (costly) local public good. More precisely, an agent chooses a i 2 f0; 1g, where a i = 1 (0) is interpreted as the decision of (not) providing a public good. Let a = (a 1 ; :::; a n ) 2 f0; 1g n denote an action proÖle. An agent providing the public good (a i = 1) pays a cost of C > 0, whereas an agent enjoying the public good (from at least one other agent, or herself) obtains a beneÖt of A > 0, where A > C. Let U i (a i ; a !i ) be the utility function of agent i given the action proÖle a = (a i ; a !i ), where a !i stands for the action chosen by all agents except for i.

Then:
There are at least two key assumptions implicit in this speciÖcation. On the one hand, there is no congestion, and thus the beneÖt of observing the public good is independent on how many other agents also beneÖt from it (i.e., the good is non-rival). On the second hand, there is no extra beneÖt derived from observing the public good more than once. 2 We can now analyze the Nash equilibria of the induced game, where all agents decide simultaneously whether or not to provide the public good. An agent would decide to provide the public good if and only if nobody observed by her is doing so. Formally, one can characterize the equilibrium as follows: a strategy proÖle (a & 1 ; a & 2 ; :::; a & n ) is a Nash equilibrium if and only if all agents providing the public good, i.e., fi 2 N; s.t. a ! i = 1g, form a maximal independent set. A maximal independent set is a set of agents satisfying that no agent observes any other agent in the set, and all agents out of the set observe at least one agent within the set. 3 Notice that, this model has little predictive power since for many complex networks there will be a large set of possible equilibrium outcomes. 4 In addition to this problem, it is easy to construct simple networks where there exists no maximal independent set, and thus the issue of non-existence of (pure strategy) Nash equilibria arises. 5 This motivates the study of a dynamic approach for which there always exists a unique prediction.

The dynamic model
We consider a dynamic model to describe the evolution of agentsí choices through time. We di §er from the static approach in several directions. On the one hand, agents update their decision to provide the public good over time. On the other hand, the ináuence network is going to change (non-strategically) every period due to the randomness assumed in the linking process. Agents are characterized by their out-degree d i , representing for instance, exogenous time constraints, where P (d) denotes the out-degree distribution in the population. At each time step the network is randomly generated given two primitives of the process that remain Öxed: (a) the degree distribution P (d) and (b) a parameter ' 2 R which introduces linking biases with respect to the out-degree. In particular, the probability that an agent links with (or observes) an agent with out-degree d is equal to The expected in-degree of an agent is deÖned as the expected number of agents that observe this agent. It is straightforward to show the expected in-degree (d in ) of 6 With some abuse of terminology, throughout the paper, the expected in-degree will simply be referred to as the in-degree. Two special cases could be singled out regarding ". If " = 0 agents observe others uniformly at random and as a consequence the out-degree and in-degree are uncorrelated, in fact, all agents have the same in-degree or visibility (see LÛpez-Pintado, 2013 for the analysis of such extreme case). If, instead, " = 1 then an agent with twice the out-degree of another agent is selected twice as often. This reáects the idea that agents that have a higher out-degree are also more visible for others and therefore will have a higher in-degree. Hereafter the in/outdegree correlation can be roughly identiÖed with parameter " when " 2 [0; 1] as an increase in such parameter corresponds with an increase in the similarity between the out-degree and in-degree of agents.
Notice that, if " > 1 agents with a high out-degree have an even higher in-degree, whereas " < 0 implies that agents with a high out-degree will typically have a very low in-degree. For most of the paper we will concentrate on the case " 2 [0; 1], although we will point out which results can be extended to other values of ".
Consider a continuous-time dynamics to describe the evolution of the provision of the local public good in the population. At each time t an agent, say i, revises his action a i at a rate ) % 0. This agent decides whether or not to contribute to the public good given the behavior of those agents observed by agent i in the ináuence network realized at time t, i.e., applying a myopic best response.
Let * d (t) denote the proportion of agents with out-degree d that are choosing action 1 at time t. A state is determined by the proÖle f* d (t)g d#1 , where we assume that all agents have at least out-degree 1. There are two measures that will be of particular importance for our analysis.
First, the overall fraction of agents choosing 1 (non-conditional on degree). This measure is denoted by *(t) and can be computed as follows: Second, the probability that a link reaches an agent choosing 1. This probability is repre- , where the numerator is the total number of links that would reach the set of agents with out-degree d and the denominator is the number of agents with out-degree d in the population.
sented by !(t) and can be computed as follows: ( 1 ) The computation of !(t) is derived from the fact that d ! P (d) hd ! i is the probability of observing an agent with out-degree d and, conditional on having out-degree d, the probability of providing the public good is % d (t). Notice that, in the extreme case where & = 0 (i.e., in/out-degree correlation is zero) then !(t) = %(t) but, in general, these two measures will di §er.
For each degree d, the deterministic approximation of the evolution of % d (t) is given by the following di §erential equation: Notice that the positive term in the equation accounts for transitions from action 0 to 1, whereas the negative term accounts for the reverse transitions (i.e., from action 1 to 0). The Örst term can be interpreted as follows: After simpliÖcations of equation (2) we Önd that, for each d, Given this system of di §erential equations we can now address the issue of equilibrium.
Therefore, given equation (1), ! $ must be a solution of the following (Öxed-point) equation: where we deÖne Once we know " " in equilibrium we can also determine % " d for each d and, consequently, the overall fraction of public good contributors % " .
Notice that, in the dynamics described above, agents can switch actions (from 0 to 1 and vice-versa) in equilibrium, given that the ináuence network is randomly generated every period.
Therefore, the concept of stationary state, only refers to stationary values for ", f% d g d and %.
The next result addresses the issue of existence and uniqueness of the equilibrium. Proof. Note that . Therefore, H P;# (") is a (continuous and) decreasing function of ". Furthermore, H P;# (0) = 1 and H P;# (1) = 0. Thus, there exists a unique solution " " 2 (0; 1) of equation (5) and therefore a unique value for " (and %) in equilibrium. To conclude, let us show that " " is globally stable, i.e., starting from any initial fraction of agents choosing 1 (% = % 0 ), the dynamics converges to a state where " = " " . To do so notice that and, substituting d' d (t) dt for its value determined by (3), we obtain that from where the desired conclusion follows.
In the next section we develop comparative statics results with respect to the in/out-degree correlation & and the out-degree distribution P (the two primitives of the ináuence network).

&.
Consider Örst the comparative statics with respect to &.
Notice that an increase in " leads to opposite e §ects on $ ! and % ! , something which, at Örst, is quite counter-intuitive. The intuition for such a result is the following. It is always the case that, in equilibrium, the fraction of agents contributing to the public good decreases with respect to the out-degree (i.e., % ! d is decreasing with respect to d) since the probability of observing an agent providing the public good is lower for agents with smaller out-degrees (see equation 4). Thus, if " increases, agents with low out-degree are observed by relatively fewer agents which implies that the probability of observing through one of the links the public good (i.e., $ ! ) would also decrease. As a consequence, to compensate for such a decrease in $ ! , the fraction of agents contributing to the public good (i.e., % ! ) increases as " increases.
The formal proof is the following.

Proof. Recall that the Öxed point equation characterizing
We can interpret Q ";P (d) as the out-degree distribution of observed agents. We show next that Q " 2 ;P (d) First Order Stochastic Dominates Q " 1 ;P (d) if and only if " 1 " " 2 . Intuitively, this should hold as a higher in/out-degree correlation implies that agents with high out-degree are observed more often which also implies that the out-degree distribution of observed agents must take larger values. Formally, we must Önd that the cumulative distribution function of Q " 2 ;P (d) is always below the cumulative distribution function of or, analogously, that Condition (6) can be written as follows: Note that the two expressions coincide, as long as d 2 is bounded below D. That is, we know For the part of the sum where d 2 exceeds D, however, this is no longer the case (since a permutation of the indices no longer appears in the sum). For such cases, as d 1 < d 2 and which proves condition (6).
To complete the proof we use that & d = (1 ! ') d is decreasing as a function of d (for all and thus that ' " (P; Notice that the result is true for all possible values of % 2 R, not just % 2 [0; 1].
In order to present the next result let us deÖne Örst the meaning of a free rider in this context. An agent is a free rider if it observe the public good but it does not provide it herself.
The next result indicates how the fraction of free riders depends on the parameter %. Proof. Notice that the fraction of free-riders, denoted by y, is equal to is the probability that an agent with out-degree d has of enjoying the public good. In equilibrium & " d = (1 ! ' " ) d and therefore which is increasing with respect to ' " . Notice that, due to Proposition 2, ' " decreases with % which proofs the result.
Consider now the comparative statics with respect to P . To do so, we consider a Öx value of % 2 [0; 1] and analyze how di §erent out-degree distributions lead to di §erent outcomes. We Örst study the e §ect of a First Order Stochastic Dominance shift of the out-degree distribution, and then analyze the e §ect of a Mean Preserving Spread.
Notice that condition (7) is satisÖed as long as d m is su¢ciently high. Roughly speaking, Proposition 3 implies that if the network becomes denser then, in equilibrium, the fraction of links reaching the public good decreases. This result also implies that ' # d (P ; !) & ' # d (P; !) for all d. Comparative statics, however, with respect to ' # might depend on further properties of the out-degree distributions. 7 The proof of the result is provided next.
which is negative if and only if Let & m be such that .
That is, ! m = 1 ! e !! dm . We can easily check that !" ln(1!#) is a decreasing function of ! which then implies that g(d) is decreasing for all d " d m , provided that ! " ! m . Therefore, as P (d) First Order Stochastic Dominates P (d), for all ! 2 [! m ; 1]. In addition, as d " is an increasing function of d we know that hd " i P " hd " i P . Thus, To complete the proof we must show that ! m # ! # (P ; (), which is the case if ! m # H P ;" (! m ), or, analogously, if the next condition holds: It is straightforward to show that Proposition 3 holds for all values of ( > 0, but does not apply to the case ( < 0, as d " would be an increasing function of d and thus hd " i P # hd " i P .
Finally we compare two out-degree distributions where one is a Mean Preserving Spread of the other one. In particular these two distribution have the same average out-degree, but di §erent variance. Which case would lead to a larger contribution in equilibrium? The next result partially addresses this question.
Note that condition (8) holds as long as d m is su¢ciently high. Proposition 4 indicates that the number of links reaching the public good (i.e., ! # ) increases with the heterogeneity of the network. This result also implies that + # d (P ; () # + # d (P; () for all d. Again, comparative statics with respect to + might depend on further properties of the out-degree distributions.
for all ! 2 [! m ; 1]. In addition, we know that hd $ i P # hd $ i P , as d $ is a concave function of d.

Thus,
To complete the proof we must show that ! m % ! % (P; '), which is the case if ! m % H P;$ (! m ) or, analogously, if the next condition holds: It is straightforward to show that Proposition 4 does not apply to the cases ' < 0 nor 1 < ', since d $ would be a convex function of d and thus hd $ i P % hd $ i P .

E¢ciency
In this section we focus on e¢ciency. We study the simplest possible version of utility aggregation which is utilitarianism. That is, we say that a state is e¢cient if it maximizes the sum of the utilities of all the agents in the population. A state in this context is characterized by the vector f! d g d!1 . In particular, let welfare be deÖned (when normalized by the population size n) as: is the fraction of agents that do not enjoy the public good. Notice that, this is analogous to the computation of the expected utility of an agent chosen uniformly at random from the population given f! d g d!1 .
To simplify matters we focus Örst on the homogeneous case where all agents have the same out-degree d. By deÖnition + plays no role in such an homogenous framework. In particular, in this case the fraction of links pointing to a public good coincides with the fraction of agents contributing to the public good, i.e., * = !. We obtain the following result.
Proposition 6 If all agents have a degree equal to d then the fraction of agents that contribute to the public good in the e¢cient state is Moreover, in equilibrium there is underprovision of the public good, that is, ! " < ! e .
The proof comes next.
Proof. We must solve the following maximization problem The second order condition is satisÖed and the Örst order condition (W 0 (!) = 0) provides the . Moreover, the equilibrium value of ! should satisfy the Öxed point equation It is straightforward to see that H(! e ) < ! e which implies, as H is decreasing, that ! " < ! e .
The previous result illustrates that the e¢cient and equilibrium outcomes do not typically coincide. It also shows that the e¢cient value of ! e increases with respect to the revenue/cost ratio A C of the public good. Thus, as A C increases the tension between e¢ciency and equilibrium is augmented. 8 The computation of the e¢cient state in a more general setting can be quite cumbersome. We present next the case of a population with two types of agents; agents with high out-degree d and agents with low out-degree d to show that not only the e¢cient and equilibrium outcomes do not coincide, but also that they exhibit distinctive properties. For simplicity let us assume that an agent can have high or low out-degree with equal probability.
That is, We show the following.
Proof. In this case welfare is equal to or analogously, and D = d(1 ! +) d"1 d

Discussion
In this paper we propose a stylized model of public good provision in a random network context.
The ináuence network is characterized by its degree distribution and the correlation between agentsí out-degree (observation level) and in-degree (visibility level). In particular, because nowadays personal interaction and ináuence is being substituted by online social networks it seems reasonable to assume that such correlations are not trivial. We Önd that, in equilibrium, an increase in the in/out-degree correlation increases the number of public good providers (i.e., !), but, on the contrary, it decreases the number of links that reach a public good provider (i.e., "). Moreover, the number of free riders decreases with in/out-degree correlations.
We have also analyzed the e §ect that a variation of the out-degree distribution has on the equilibrium outcomes. Our results in this respect show that, if degrees are su¢ciently large, an increase in the average level of information (i.e., an increase in the average out-degree) decreases the fraction of links reaching the public good, whereas an increase in the dispersion of information (i.e., an increase in the variance of P ) increases such fraction.
Finally we show that there is misalignment between the e¢cient outcome and the equilibrium outcome which becomes more important as the revenue/cost ratio of the public good (A=C) increases (at least in the homogeneous case). We illustrate an additional tension between e¢cient and equilibrium states; the probability of contributing to the public good decreases with respect to out-degree in equilibrium, whereas it increases in the e¢cient state.
This paper contributes to the growing literature on public goods in networks by bridging the work developed in statistical physics (where random networks are commonly used) with the literature in economics, for which the problem of public good provision is a major topic of study. The assumption that the network is randomly generated every period is quite strong and thus, one possible direction for further study would be to enrich our model by allowing for clustering and community structures. This extension has already been addressed for contagion models with heterogenous agents and homophily providing fruitful results (e.g., Jackson and LÛpez-Pintado, 2013).